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Axiom ax-4 1334
Description: Axiom of Specialization. A quantified wff implies the wff without a quantifier (i.e. an instance, or special case, of the generalized wff). In other words if something is true for all x, it is true for any specific x (that would typically occur as a free variable in the wff substituted for φ). (A free variable is one that does not occur in the scope of a quantifier: x and y are both free in x = y, but only x is free in yx = y.) Axiom scheme C5' in [Megill] p. 448 (p. 16 of the preprint). Also appears as Axiom B5 of [Tarski] p. 67 (under his system S2, defined in the last paragraph on p. 77).

Note that the converse of this axiom does not hold in general, but a weaker inference form of the converse holds and is expressed as rule ax-gen 1270. Conditional forms of the converse are given by ax-12 1336, ax-16 1605, and ax-17 1352.

Unlike the more general textbook Axiom of Specialization, we cannot choose a variable different from x for the special case. For use, that requires the assistance of equality axioms, and we deal with it later after we introduce the definition of proper substitution - see stdpc4 1569.

(Contributed by NM, 5-Aug-1993.)

Ref Expression
ax-4 (xφφ)

Detailed syntax breakdown of Axiom ax-4
StepHypRef Expression
1 wph . . 3 wff φ
2 vx . . 3 set x
31, 2wal 1267 . 2 wff xφ
43, 1wi 4 1 wff (xφφ)
Colors of variables: wff set class
This axiom is referenced by:  sp  1335  ax-12  1336  hbequid  1340  a4i  1363  hbim  1371  19.3h  1379  19.21h  1383  19.21bi  1386  hbimd  1398  19.21ht  1406  hbnt  1465  19.12  1476  19.38  1484  ax9o  1502  hbae  1519  equveli  1552  sb2  1561  drex1  1589  ax11b  1617  sbf3t  1638  a16gb  1654  sb56  1673  sb6  1674  sbalyz  1781  hbsb4t  1793  mopick  2573  2eu1  2589  dfsb2  2621
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